Unless I've climbed above a tight crotch or branch I have to worry about pulling my link back through I attach my pull down rope on the loose end of the rope right next to the link. Doing it as shown in the diagram on the right below greatly eases the force necessary to pull the girth hitch loose since you don't have to defeat the friction of the rope passing all the way around the tree. The down side is that this can leave the link in the tree which can make it difficult to pull through a tight crevice if there is one. You can pull the link down first if you have enough rope by pulling the rope down until you can reach the loop you're pulling then drop the pull down rope and grab the tag end and the loop you just pulled down. Pull the tag end while loosely holding the loop and letting the rope slide back around the tree and the link will come down out of the tree first.
Honestly though, if I've climbed over a tight crevice I hook up right behind the link on the backside of the tree and pull the link out directly. If it hangs up at all I find generally backing away from the tree a bit and then alternating pressure between the tag end of the rope and the pull down rope will work it loose pretty quick.
Example shown below:
Blue line = Climbing Rope, Grey line = Pull down string, Red Arrows = where rope has to move to come loose
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